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Black holes in 4d $\mathcal{N}=4$ Super-Yang-Mills

Francesco Benini, Elisa Milan

TL;DR

This work demonstrates that large-$N$ $ ext{N}=4$ SU$(N)$ SYM on $S^3$ houses enough 1/16-BPS states to account for the entropy of rotating AdS$_5$ black holes, by leveraging a Bethe Ansatz formulation of the superconformal index. The authors identify an exponentially large sector in the index at generic complex fugacities that reproduces the Bekenstein-Hawking entropy for Gutowski-Reall black holes via a Legendre transform, while revealing a rich structure of competing exponentials and Stokes lines that signal new physics beyond a single saddle. They also relate the index analysis to $ ext{I}$-extremization and an entropy function framework, and discuss potential instabilities toward hairy or multi-center configurations in certain charge regimes. Overall, the paper provides a nonperturbative CFT count that matches gravitational entropy in AdS$_5$ and uncovers intricate large-$N$ analytic structure with implications for black hole microphysics and holography.

Abstract

We resolve a long-standing question: does the four-dimensional $\mathcal{N}=4$ SU(N) Super-Yang-Mills theory on $S^3$ at large N contain enough states to account for the entropy of rotating electrically-charged BPS black holes in AdS$_5$? Our answer is positive. We reconsider the large N limit of the superconformal index, using the Bethe Ansatz formulation, and find an exponentially large contribution which exactly reproduces the Bekenstein-Hawking entropy of the black holes of Gutowski-Reall. Besides, the large N limit exhibits a complicated structure, with many competing exponential contributions and Stokes lines, hinting at new physics.

Black holes in 4d $\mathcal{N}=4$ Super-Yang-Mills

TL;DR

This work demonstrates that large- SU SYM on houses enough 1/16-BPS states to account for the entropy of rotating AdS black holes, by leveraging a Bethe Ansatz formulation of the superconformal index. The authors identify an exponentially large sector in the index at generic complex fugacities that reproduces the Bekenstein-Hawking entropy for Gutowski-Reall black holes via a Legendre transform, while revealing a rich structure of competing exponentials and Stokes lines that signal new physics beyond a single saddle. They also relate the index analysis to -extremization and an entropy function framework, and discuss potential instabilities toward hairy or multi-center configurations in certain charge regimes. Overall, the paper provides a nonperturbative CFT count that matches gravitational entropy in AdS and uncovers intricate large- analytic structure with implications for black hole microphysics and holography.

Abstract

We resolve a long-standing question: does the four-dimensional SU(N) Super-Yang-Mills theory on at large N contain enough states to account for the entropy of rotating electrically-charged BPS black holes in AdS? Our answer is positive. We reconsider the large N limit of the superconformal index, using the Bethe Ansatz formulation, and find an exponentially large contribution which exactly reproduces the Bekenstein-Hawking entropy of the black holes of Gutowski-Reall. Besides, the large N limit exhibits a complicated structure, with many competing exponential contributions and Stokes lines, hinting at new physics.

Paper Structure

This paper contains 31 sections, 197 equations, 4 figures.

Table of Contents

  1. Introduction and results
  2. BPS black holes in AdS$\boldsymbol{_5}$
  3. The dual field theory and its index
  4. Exact solutions to the BAEs
  5. The large $\boldsymbol{N}$ limit
  6. Contribution of the basic solution
  7. The Jacobian.
  8. The functions $\boldsymbol{\widetilde{\Gamma}}$.
  9. Total contribution from the basic solution.
  10. Contribution of $\boldsymbol{SL(2,\mathbb{Z})}$-transformed solutions
  11. Final result and Stokes lines
  12. Comparison with previous literature
  13. Statistical interpretation and $\boldsymbol{\mathcal{I}}$-extremization
  14. Black hole entropy from the index
  15. The entropy function.
  16. ...and 16 more sections

Figures (4)

  • Figure 1: In yellow is highlighted the domain (\ref{['regime of convergence']}) in the complex $\Delta$-plane. The right boundary is the line $\gamma$, passing through 0 and $\tau$. The left boundary is the line $\gamma - 1$, passing through $-1$ and $\tau - 1$. The dashes lines are other elements of $\gamma + \mathbb{Z}$.
  • Figure 2: The upper left plot shows the values of $\mathop{\mathrm{\mathbb{I}m}}\nolimits\Theta(\Delta;\tau + r)$ as a function of $r$, for $\tau$ inside the semi-circle \ref{['red circle']}. The red dot corresponds to $r=0$, and is the dominant contribution in this case. The upper right plot shows $\mathop{\mathrm{\mathbb{I}m}}\nolimits\Theta(\Delta;\tau + r)$ for $\tau$ inside the semi-circle \ref{['green circle']}. The green dot corresponds to $r=-1$, and is the dominant contribution in this case. The lower plot shows the values of $\mathop{\mathrm{\mathbb{I}m}}\nolimits\Theta(\Delta;\tau + r)$ for $\tau$ outside the two semicircles, where there is no dominant contribution.
  • Figure 3: Stokes lines in the $\tau$-plane. The red semicircle corresponds to the domain of analyticity where the $r=0$ contribution dominates the index. The green semicircle instead corresponds to the domain where the $r=-1$ contribution dominates. In the remaining region we do not have a dominant contribution. The blue and orange lines are the critical points of the entropy function for BPS black holes, in the two possible formulations. The dashed lines indicate the subspace where both fugacities $q$ and $y$ are real and the computation of Kinney:2005ej applies.
  • Figure 4: Stokes lines in the $q$-plane, where the variable is $q^{1/3}$. The notation is the same as in Figure \ref{['fig: tau']}.